Computing the spectrum of the Schrödinger operator \(-\Delta - \kappa /r\)  on \(B(R)\)  and \(\mathbb {R}^n\)  is closely related to computing the spectrum of the hydrogen atom from quantum mechanics. In fact, on \(\mathbb {R}^3\)  this calculation is precisely the hydrogen atom calculation (see the book by Landau and Lifschitz [1]}, for example). For general \(n \ge 3\) , the ODE in (REF ) below is the radial part of the Schrödinger operator for the 3-D hydrogen atom with angular moment \(m\)  (after changing variables). This physical interpretation breaks down when \(n =2\)  since in this case \(m < 0\) . The operator in (REF ) also appears in work of Khalile and Pankrashkin [2]} as an effective operator for the Robin Laplacian with negative Robin parameter on \({\theta }\)  as \(\theta \rightarrow 0\) , when \(n=2\) .
